Suppose Г is a group acting on a set X. An r-labeling f : X -+ { I, 2, ... , r} of X is distinguishing (with respect to Г) if the only label preserving permutation of X in Г is the identity. The distinguishing number, Dr(X), of the action of Г on X is the minimum r for which there is an r-labeling which is distinguishing. This paper investigates the relation between the cardinality of a set X and the distinguishing numbers of group actions on X. For a positive integer n, let D(n) be the set of distinguishing numbers of transitive group actions on a set X of cardinality n, i.e., D(n) = {DГ(X) : |X| = n and Г acts transitively on X}. We prove that |D(n)| = O( ~). Then we consider the problem of an arbitrary fixed group Г acting on a large set. We prove that if for any action of Г on a set Y, for each proper normal subgroup H of F, D y (Y) ≤ 2, then there is an integer a such that for any set X with |X| ≥ n, for any action of Г on X with no fixed points, DГ(X) ≤ 2.
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机译：假设Г是作用在集合X上的一个基团。如果唯一的保留X的置换在X中，则X的r标记f：X-+ {I，2，...，r}可以区分（关于Г）。 Г是身份。 Г在X上的作用的区分数Dr（X）是最小r，在该最小r上有区分的r-标记。本文研究了集合X的基数与X上的组动作的可区分数目之间的关系。对于正整数n，令D（n）是基数为n的集合X上的传递性群组动作的可区分数目的集合，即D（n）= {DГ（X）：| X | = n，且Г对X}具有传递作用。我们证明| D（n）| = O（〜）。然后，我们考虑一个任意固定组Г作用于大集合的问题。我们证明如果对于Г对集合Y的任何作用，对于F的每个适当的正常子群H，D y（Y）≤2，那么将存在一个整数a，使得对于具有| X |的任何集合X。 ≥n，对于Г在X上没有固定点的任何作用，DГ（X）≤2。
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